Problem 44 - Chemistry: Atoms First

Question

Express each of the following numbers in scientific notation with correct significant figures: (a) 711.0 (b) 0.239 (c) 90743 (d) 134.2 (e) 0.05499 (f) 10000.0 (g) 0.000000738592

Step-by-Step Solution

Answer

\(711.0 = 7.110 \times 10^{2}\), \(0.239 = 2.39 \times 10^{-1}\), \(90743 = 9.0743 \times 10^{4}\), \(134.2 = 1.342 \times 10^{2}\), \(0.05499 = 5.499 \times 10^{-2}\), \(10000.0 = 1.00000 \times 10^{4}\), \(0.000000738592 = 7.38592 \times 10^{-7}\).

Step 1: Identify Significant Figures for 711.0

Start by identifying the significant figures in the number 711.0. There are four significant figures: the digits 7, 1, 1, and the last 0, since it comes after the decimal point.

Step 2: Convert 711.0 to Scientific Notation

Move the decimal point so that there is one non-zero digit to the left of the decimal point. The number 711.0 becomes 7.110. Then, count the number of places the decimal has moved, which is 2. Write the number in scientific notation as: \( 7.110 \times 10^{2} \).

Step 3: Identify Significant Figures for 0.239

Identify significant figures in the number 0.239. There are three significant figures: the digits 2, 3, and 9.

Step 4: Convert 0.239 to Scientific Notation

Move the decimal point in the number 0.239 to get one non-zero digit to the left of the decimal; the result is 2.39. The decimal point moved 1 place to the right, so the scientific notation is: \( 2.39 \times 10^{-1} \).

Step 5: Identify Significant Figures for 90743

Identify the significant figures in the number 90743. All five digits are significant because they are non-zero.

Step 6: Convert 90743 to Scientific Notation

The number 90743 will have the decimal moved 4 places to the left to become 9.0743. Therefore, the scientific notation is: \( 9.0743 \times 10^{4} \).

Step 7: Identify Significant Figures for 134.2

Identify the significant figures in the number 134.2. There are four significant figures: 1, 3, 4, and 2.

Step 8: Convert 134.2 to Scientific Notation

Move the decimal 2 places to the left to make 1.342. The scientific notation is: \( 1.342 \times 10^{2} \).

Step 9: Identify Significant Figures for 0.05499

The number 0.05499 has four significant figures: 5, 4, 9, and 9.

Step 10: Convert 0.05499 to Scientific Notation

Shift the decimal point 2 places to the right to have one non-zero digit on the left, resulting in 5.499. Therefore, the scientific notation is: \( 5.499 \times 10^{-2} \).

Step 11: Identify Significant Figures for 10000.0

The number 10000.0 has six significant figures because the trailing zero is after the decimal point.

Step 12: Convert 10000.0 to Scientific Notation

Shift the decimal point 4 places to the left which gives us 1.00000. This results in the scientific notation: \( 1.00000 \times 10^{4} \).

Step 13: Identify Significant Figures for 0.000000738592

Identify significant figures in 0.000000738592. All the non-zero digits are significant, totaling eight significant figures.

Step 14: Convert 0.000000738592 to Scientific Notation

Move the decimal point 7 places to the right to yield 7.38592. Thus, the scientific notation is: \( 7.38592 \times 10^{-7} \).

Key Concepts

Significant Figures

Understanding significant figures is crucial in science because they reflect the precision of a measurement or calculation. Significant figures are the digits in a number that are known with certainty plus one last digit, which is somewhat uncertain or estimated. When determining significant figures, we follow certain rules:

• Non-zero digits are always significant.
• Any zeros between non-zero digits are significant.
• Leading zeros (zeros before non-zero numbers) are not significant.
• Trailing zeros in a number with a decimal point are significant.

The importance of significant figures becomes clear when performing arithmetic operations, as the result should only be as precise as the least precise measurement used in the calculation. For example, in the given problem, the number 711.0 has four significant figures. The trailing zero is significant because it's after the decimal point, showing precision up to the tenths place. In scientific notation, we maintain all significant figures to accurately represent the original number.

Expressing Numbers in Scientific Notation

Scientific notation is a method used to express very large or small numbers in a compact form. It is especially useful in science and engineering to handle extreme values conveniently. The general format for scientific notation is: \( a \times 10^n \), where \( n \) is an integer, and \( a \) is a number with absolute value between 1 and 10 that includes all of the significant figures.
For instance, in our exercise, expressing the number 90743 in scientific notation involves placing the decimal after the first significant digit and noting the number of places the decimal has moved. Hence, 90743 becomes \( 9.0743 \times 10^4 \).
This approach simplifies calculations and maintains the precision of the numbers involved. It's especially useful when working with numbers that originate from measurements, as these often have a set level of precision and need to be compared or combined with similar precision.

Decimal Point Movement

The process of converting a number to scientific notation involves moving the decimal point to a new position. This is essential for expressing the number with one non-zero digit to the left of the decimal. The direction and number of places you move the decimal point become the power of ten in the scientific notation:

• Move to the left for numbers greater than 10; the exponent will be positive.
• Move to the right for numbers less than 1; the exponent will be negative.

For example, to convert the number 0.05499 to scientific notation, as shown in the solution, you move the decimal two places to the right, resulting in 5.499, which becomes \( 5.499 \times 10^{-2} \) in scientific notation. The exponent, -2, indicates the decimal was moved two places to the right. Decimal point movement is a visualization of the scale of the number and it's an important skill for expressing numbers in scientific notation properly.